Diffusion models have been firmly established as principled zero-shot solvers for linear and nonlinear inverse problems, owing to their powerful image prior and iterative sampling algorithm. These approaches often rely on Tweedie's formula, which relates the diffusion variate $\mathbf{x}_t$ to the posterior mean $\mathbb{E} [\mathbf{x}_0 | \mathbf{x}_t]$, in order to guide the diffusion trajectory with an estimate of the final denoised sample $\mathbf{x}_0$. However, this does not consider information from the measurement $\mathbf{y}$, which must then be integrated downstream. In this work, we propose to estimate the conditional posterior mean $\mathbb{E} [\mathbf{x}_0 | \mathbf{x}_t, \mathbf{y}]$, which can be formulated as the solution to a lightweight, single-parameter maximum likelihood estimation problem. The resulting prediction can be integrated into any standard sampler, resulting in a fast and memory-efficient inverse solver. Our optimizer is amenable to a noise-aware likelihood-based stopping criteria that is robust to measurement noise in $\mathbf{y}$. We demonstrate comparable or improved performance against a wide selection of contemporary inverse solvers across multiple datasets and tasks.

翻译：扩散模型凭借其强大的图像先验和迭代采样算法，已被确立为线性和非线性逆问题的原理性零样本求解器。这类方法通常依赖Tweedie公式——该公式将扩散变量$\mathbf{x}_t$与后验均值$\mathbb{E} [\mathbf{x}_0 | \mathbf{x}_t]$相关联——从而通过最终去噪样本$\mathbf{x}_0$的估计值引导扩散轨迹。然而，该方法未考虑测量值$\mathbf{y}$中的信息，导致需在后续阶段整合该信息。本文提出估计条件后验均值$\mathbb{E} [\mathbf{x}_0 | \mathbf{x}_t, \mathbf{y}]$，该估计可转化为轻量级单参数极大似然估计问题的解。所得预测结果可集成至任意标准采样器，形成快速且内存高效的逆问题求解器。我们提出的优化器兼容基于噪声感知的似然终止准则，对$\mathbf{y}$中的测量噪声具有鲁棒性。在多个数据集和任务中，该方法与当代多种逆问题求解器相比，展现出相当或更优的性能。